Why does the shower curtain move towards me when I am taking a hot shower?

It is due to the convection in the shower, as mentioned in Steeven's comment.

The water is significantly above room temperature. Obviously it is going to heat the air (and you can feel the warm air/vapour yourself).

The air in the shower is both warming up, and becoming more humid. Both of those factors lower the density of the air, making it raise up.

When the air flows up, it creates a low pressure zone in the shower. Once the pressure gets low enough compared to the room outside, the pressure difference pushes the curtain into the shower until enough air can get in to account for the air lost to convection.

Warmer water will cause greater convective flows, so the effect should be more noticeable.

Disclaimer: Note that the heat and humidity are not the only cause of convective flows; and convective airflow is likely only part of the situation. As some comments (and the wikipedia page on this subject) mentioned, the flow of the water downwards may actually generate an air vortex in the shower. This vortex creates a low pressure zone, which drives the convective process regardless of water temperature.

There are also other potential reasons, and without experiments and potentially (multi-physics) simulations, it's very unclear what the primary cause of this phenomenon is.

I still feel this answer properly addresses the second question about why warmer water makes the process more noticeable.

EDIT on reflection, my answer, as noted in the comments below, is not correct (to be specific, it's dead in the water), it simply would not produce the mass of less dense air required. END EDIT

The Bernoulli Principle is probably the main cause. This states that an increase in velocity, of the air associated with the falling water, produces a reduction in air pressure around your body, and then ambient air pressure in the bathroom pushes the curtain towards you.

Another possible reason is the Coanda Effect, which you can observe when you hold a spoon near a stream of water from a tap and as the water moves over the curved back of the spoon, the spoon will move inwards toward jet center.

I have to say I include this Coanda Effect for interest only, I believe the first reason to be far more significant.

There are numerous theories and this has been a subject of eternal debate. However, given that we now have mountains of evidence, I believe at this point we can state conclusively that the mechanism is as follows:

The water falls in the front of the shower. The falling water carries air downward with it. The air that moves downward at the very front of the shower moves upward further back. This pattern of downward moving air in the front and upward moving air in the back closes to form a stable, horizontal vortex. The vortex has a low-pressure zone in the middle, which produces a constant, low-pressure against part of the curtain. The higher air pressure on the other side of the air curtain therefore pushes it inward.